The Dynamics of Vector-Host Feeding Contact Rate with Saturation: A Case of Malaria in Western Kenya

ABSTRACT

In this study, we develop an expression for a saturated mosquito feeding rate in an SIS malaria model to determine its effect on infection and transmission dynamics of malaria in the highlands of Western Kenya. The basic reproduction number is established as a sharp threshold that determines whether the disease dies out or persists in the population. Precisely, if , the disease-free equilibrium is globally asymptotically stable and the disease always dies out and if , there exists a unique endemic equilibrium which is globally stable and the disease persists. The contribution of the saturated contact rate to the basic reproduction number and the level of the endemic equilibrium are also analyzed.

Cite this paper

Wairimu, J. and Wandera, O. (2013) The Dynamics of Vector-Host Feeding Contact Rate with Saturation: A Case of Malaria in Western Kenya.*Applied Mathematics*, **4**, 1381-1391. doi: 10.4236/am.2013.410187.

Wairimu, J. and Wandera, O. (2013) The Dynamics of Vector-Host Feeding Contact Rate with Saturation: A Case of Malaria in Western Kenya.

References

[1] S. I. Hay, M. Simba, M.Busolo, A. M. Noor, H. L. Guyatt, S. A. Ochola and R. W. Snow, “Defining and Detecting Malaria Epidemics in the Highlands of Western Kenya,” Emerging Infectious Diseases, Vol. 8, No. 6, 2002, pp. 555-562.

[2] B. Ndenga, A. Githeko, E. Omukunda, G. Munyekenye, H. Atieli, P. Wamai, C. Mbogo, N. Minakawa, G. Zhou and G. Yan, “Population Dynamics of Malaria Vectors in Western Kenya Highlands,” Journal of Medical Ento mology, Vol. 43, No. 2, 2006, pp. 200-206.

[3] R. E. Gurtler, L. A. Ceballos, P. O. Krasnowski, L. A. Lanati and R. Stariolo, “Strong Host-Feeding Preferences of the Vector Triatoma Infestans Modified by Vector Density: Implications for the Epidemiology of Chagas Disease,” PLoS Neglected Tropical Diseases, Vol. 3, No. 5, 2009, p. e447.

[4] C. S. Holling, “The Functional Response of Predators to Prey Density and Its Role in Mimicry and Population Regulations,” Memoirs of the Entomological Society of Canada, Vol. 97, No. S45, 1965, pp. 5-60.

[5] M. J. Klowden, “The Endogenous Regulation of Mosquito Reproductive Behavior,” Experientia, Vol. 46, No. 7, 1990, pp. 660-670.

[6] G. A. Ngwa, “On the Population Dynamics of the Malaria Vector,” Bulletin of Mathematical Biology, Vol. 68, No. 8, 2006, pp. 2161-2189.

[7] D. W. Kelly and C. E. Thompson, “Epidemiology and Optimal Foraging: Modelling the Ideal Free Distribution of Insect Vectors,” Parasitology, Vol. 120, No. 3, 2000, pp. 319-327.

[8] F. J. Lopez-Antunano, “Epidemiology and Control of Malaria and Other Arthropod-Borne Diseases,” Memórias do Instituto Oswaldo Cruz, Vol. 87, No. 3, 1992, pp. 105-114.

[9] C. Jost, O. Arino and R. Arditi, “About Deterministic Ex tinction in a Ratio-Dependent Predator-Prey Model,” Bulletin of Mathematical Biology, Vol. 61, No. 1, 1999, pp. 19-32.

[10] S. G. Staedke, E. W. Nottingham, J. C. Kamya, M. R. Rosenthal and P. J. Dorsey, “Short Report: Proximity to Mosquito Breeding Sites as a Risk Factor for Clinical Malaria Episodes in an Urban Cohort of Ugandan Chil dren,” The American Journal of Tropical Medicine and Hygiene, Vol. 69, No. 3, 2003, pp. 244-246.

[11] A. L. Menach, F. E. McKenzie, A. Flahault and D. L. Smith, “The Unexpected Importance of Mosquito Ovi position Behaviour for Malaria: Nonproductive Larval Habitats Can Be Sources for Malaria Transmission,” Ma laria Journal, Vol. 4, 2005, p. 23. http://dx.doi.org/10.1186/1475-2875-4-23

[12] C. Kribs-Zaleta, “Estimating Contact Process Saturation in Sylvatic Transmission of Trypanosoma Cruzi in the United States,” PLoS Neglected Tropical Diseases, Vol. 4, No. 4, 2010, p. e656. http://dx.doi.org/

10.1371/journal.pntd.0000656

[13] V. Capasso and G. Serio, “A Generalisation of the Kermack-Mckendrick Deterministic Epidemic Model,” Ma thematical Biosciences, Vol. 42, No. 1-2, 1978, pp. 43-61.

[14] J. A. P. Heesterbeek and J. A. J. Metz, “The Saturating Contact Rate in Marriage and Epidemic Models,” Journal of Mathematical Biology, Vol. 31, No. 5, 1993, pp. 529-539.

[15] R. Xu and Z. Ma, “Global Stability of a Sir Epidemic Model with Nonlinear Incidence Rate and Time Delay,” Nonlinear Analysis: Real World Applications, Vol. 10, No. 5, 2009, pp. 3175-3189.

[16] J. Zhang and Z. Ma, “Global Dynamics of an SEIR Epi demic Model with Saturating Contact Rate,” Mathemati cal Biosciences, Vol. 185, No. 1, 2003, pp. 15-32.

[17] L. M. Cai and X. Z. Li, “Global Analysis of a Vector Host Epidemic Model with Nonlinear Incidences,” Ap plied Mathematics and Computation, Vol. 217, No. 7, 2010, pp. 3531-3541.

[18] L. Esteva and C. Vargas, “A Model for Dengue Disease with Variable Human Population,” Journal of Mathe matical Biology, Vol. 38, No. 3, 1999, pp. 220-224.

[19] G. A. Ngwa and W. S. Shu, “A Mathematical Model for Endemic Malaria with Variable Human and Mosquito Populations,” Mathematical and Computer Modelling, Vol. 32, No. 7-8, 2000, pp. 747-763.

[20] R. Ross, “The Prevention of Malaria,” Springer-Verlag, Berlin, 1911.

[21] C. Kribs-Zaleta, “Vector Consumption and Contact Proc ess Saturation in Sylvatic Transmission of T. cruzi,” Mathematical Population Studies, Vol. 13, No. 3, 2006, pp. 135-152.

[22] K. Dietz, “Overall Population Patterns in the Transmission Cycle of Infectious Disease Agents,” Springer, Berlin, 1982.

[23] C. M. Kribs-Zaleta, “To Switch or Taper off: The Dy namics of Saturation,” Mathematical Biosciences, Vol. 192, No. 2, 2004, pp. 137-152.

[24] C. M Kribs-Zaleta, “Sharpness of Saturation in Harvest ing and Predation,” Mathematical Biosciences and Engi neering, Vol. 6, No. 4, 2009, pp. 719-742.

[25] P. Auger, E. Kouokam, G. Sallet, M. Tchuente and B. Tsanou, “The Rossmacdonald Mode in a Patchy Envi ronment,” Mathematical Biosciences, Vol. 216, No. 2, 2008, pp. 123-131.

[26] J. Tumwiine, J. Y. T. Mugisha and L. S. Luboobi, “A Mathematical Model for the Dynamics of Malaria in a Human Host and Mosquito Vector with Temporary Im munity,” Applied Mathematics and Computation, Vol. 189, No. 2, 2007, pp. 1953-1965.

[27] N. J. T. Bailey, “The Mathematical Theory of Infectious Diseases and Its Application,” Macmillan Publishers, London, 1975.

[28] H. W. Hethcote, “Qualitative Analysis of Communicable Disease Models,” Mathematical Biosciences, Vol. 28, No. 3-4, 1976, pp. 335-356.

[29] Bony and J. Michel, “Principe du Maximum, Inégalite de Harnack et Unicitédu Problème de Cauchy Pour les Opérateurs Elliptiques Dégénérés,” Annales de l’institut Fourier, Vol. 19, No. 1, 1969, pp. 277-304.

[30] M. Quincampoix, “Differential Inclusions and Target Problems,” SIAM Journal on Control and Optimization, Vol. 30, No. 2, 1992, pp. 324-335.

[31] M. Vidyasagar, “Decomposition Techniques for Large Scale Systems with Nonadditive Interactions: Stability and Stabilizability,” IEEE Transactions on Automatic Control, Vol. 25, No. 4, 1980, pp. 773-779.

[32] P. Van Den Driessche and J. Watmough, “Reproduction Numbers and Subthreshold Endemic Equilibria for Com partmental Models of Disease Transmission,” Mathe matical Biosciences, Vol. 180, No. 1-2, 2002, pp. 29-48.

[33] O. Diekmann, J. A. P. Heesterbeek and J. A. J. Metz, “On the Definition and the Computation of the Basic Repro duction Ratio R0 in Models for Infectious Diseases in Heterogeneous Populations,” Journal of Mathematical Biology, Vol. 28, No. 4, 1990, pp. 365-382.

[34] J. K. Hale, “Ordinary Differential Equations,” Krieger, 1980.

[35] J. P. LaSalle, “The Stability of Dynamical Systems,” Society for Industrial and Applied Mathematics, Phila delphia, 1976.

[36] J. P. LaSalle, “Stability Theory for Ordinary Differential Equations,” Journal of Differential Equations, Vol. 4, No. 1, 1968, pp. 57-65.

[37] J. A. Yorke, H. W. Hethcote and A. Nold, “Dynamics and Control of the Transmission of Gonorrhea,” Sexually Transmitted Diseases, Vol. 5, No. 2, 1978, pp. 51-56.

[38] C. C. McCluskey, “Global Stability for an SIR Epidemic Model with Delay and Nonlinear Incidence,” Nonlinear Analysis: Real World Applications, Vol. 11, No. 4, 2010, pp. 3106-3109.

[1] S. I. Hay, M. Simba, M.Busolo, A. M. Noor, H. L. Guyatt, S. A. Ochola and R. W. Snow, “Defining and Detecting Malaria Epidemics in the Highlands of Western Kenya,” Emerging Infectious Diseases, Vol. 8, No. 6, 2002, pp. 555-562.

[2] B. Ndenga, A. Githeko, E. Omukunda, G. Munyekenye, H. Atieli, P. Wamai, C. Mbogo, N. Minakawa, G. Zhou and G. Yan, “Population Dynamics of Malaria Vectors in Western Kenya Highlands,” Journal of Medical Ento mology, Vol. 43, No. 2, 2006, pp. 200-206.

[3] R. E. Gurtler, L. A. Ceballos, P. O. Krasnowski, L. A. Lanati and R. Stariolo, “Strong Host-Feeding Preferences of the Vector Triatoma Infestans Modified by Vector Density: Implications for the Epidemiology of Chagas Disease,” PLoS Neglected Tropical Diseases, Vol. 3, No. 5, 2009, p. e447.

[4] C. S. Holling, “The Functional Response of Predators to Prey Density and Its Role in Mimicry and Population Regulations,” Memoirs of the Entomological Society of Canada, Vol. 97, No. S45, 1965, pp. 5-60.

[5] M. J. Klowden, “The Endogenous Regulation of Mosquito Reproductive Behavior,” Experientia, Vol. 46, No. 7, 1990, pp. 660-670.

[6] G. A. Ngwa, “On the Population Dynamics of the Malaria Vector,” Bulletin of Mathematical Biology, Vol. 68, No. 8, 2006, pp. 2161-2189.

[7] D. W. Kelly and C. E. Thompson, “Epidemiology and Optimal Foraging: Modelling the Ideal Free Distribution of Insect Vectors,” Parasitology, Vol. 120, No. 3, 2000, pp. 319-327.

[8] F. J. Lopez-Antunano, “Epidemiology and Control of Malaria and Other Arthropod-Borne Diseases,” Memórias do Instituto Oswaldo Cruz, Vol. 87, No. 3, 1992, pp. 105-114.

[9] C. Jost, O. Arino and R. Arditi, “About Deterministic Ex tinction in a Ratio-Dependent Predator-Prey Model,” Bulletin of Mathematical Biology, Vol. 61, No. 1, 1999, pp. 19-32.

[10] S. G. Staedke, E. W. Nottingham, J. C. Kamya, M. R. Rosenthal and P. J. Dorsey, “Short Report: Proximity to Mosquito Breeding Sites as a Risk Factor for Clinical Malaria Episodes in an Urban Cohort of Ugandan Chil dren,” The American Journal of Tropical Medicine and Hygiene, Vol. 69, No. 3, 2003, pp. 244-246.

[11] A. L. Menach, F. E. McKenzie, A. Flahault and D. L. Smith, “The Unexpected Importance of Mosquito Ovi position Behaviour for Malaria: Nonproductive Larval Habitats Can Be Sources for Malaria Transmission,” Ma laria Journal, Vol. 4, 2005, p. 23. http://dx.doi.org/10.1186/1475-2875-4-23

[12] C. Kribs-Zaleta, “Estimating Contact Process Saturation in Sylvatic Transmission of Trypanosoma Cruzi in the United States,” PLoS Neglected Tropical Diseases, Vol. 4, No. 4, 2010, p. e656. http://dx.doi.org/

10.1371/journal.pntd.0000656

[13] V. Capasso and G. Serio, “A Generalisation of the Kermack-Mckendrick Deterministic Epidemic Model,” Ma thematical Biosciences, Vol. 42, No. 1-2, 1978, pp. 43-61.

[14] J. A. P. Heesterbeek and J. A. J. Metz, “The Saturating Contact Rate in Marriage and Epidemic Models,” Journal of Mathematical Biology, Vol. 31, No. 5, 1993, pp. 529-539.

[15] R. Xu and Z. Ma, “Global Stability of a Sir Epidemic Model with Nonlinear Incidence Rate and Time Delay,” Nonlinear Analysis: Real World Applications, Vol. 10, No. 5, 2009, pp. 3175-3189.

[16] J. Zhang and Z. Ma, “Global Dynamics of an SEIR Epi demic Model with Saturating Contact Rate,” Mathemati cal Biosciences, Vol. 185, No. 1, 2003, pp. 15-32.

[17] L. M. Cai and X. Z. Li, “Global Analysis of a Vector Host Epidemic Model with Nonlinear Incidences,” Ap plied Mathematics and Computation, Vol. 217, No. 7, 2010, pp. 3531-3541.

[18] L. Esteva and C. Vargas, “A Model for Dengue Disease with Variable Human Population,” Journal of Mathe matical Biology, Vol. 38, No. 3, 1999, pp. 220-224.

[19] G. A. Ngwa and W. S. Shu, “A Mathematical Model for Endemic Malaria with Variable Human and Mosquito Populations,” Mathematical and Computer Modelling, Vol. 32, No. 7-8, 2000, pp. 747-763.

[20] R. Ross, “The Prevention of Malaria,” Springer-Verlag, Berlin, 1911.

[21] C. Kribs-Zaleta, “Vector Consumption and Contact Proc ess Saturation in Sylvatic Transmission of T. cruzi,” Mathematical Population Studies, Vol. 13, No. 3, 2006, pp. 135-152.

[22] K. Dietz, “Overall Population Patterns in the Transmission Cycle of Infectious Disease Agents,” Springer, Berlin, 1982.

[23] C. M. Kribs-Zaleta, “To Switch or Taper off: The Dy namics of Saturation,” Mathematical Biosciences, Vol. 192, No. 2, 2004, pp. 137-152.

[24] C. M Kribs-Zaleta, “Sharpness of Saturation in Harvest ing and Predation,” Mathematical Biosciences and Engi neering, Vol. 6, No. 4, 2009, pp. 719-742.

[25] P. Auger, E. Kouokam, G. Sallet, M. Tchuente and B. Tsanou, “The Rossmacdonald Mode in a Patchy Envi ronment,” Mathematical Biosciences, Vol. 216, No. 2, 2008, pp. 123-131.

[26] J. Tumwiine, J. Y. T. Mugisha and L. S. Luboobi, “A Mathematical Model for the Dynamics of Malaria in a Human Host and Mosquito Vector with Temporary Im munity,” Applied Mathematics and Computation, Vol. 189, No. 2, 2007, pp. 1953-1965.

[27] N. J. T. Bailey, “The Mathematical Theory of Infectious Diseases and Its Application,” Macmillan Publishers, London, 1975.

[28] H. W. Hethcote, “Qualitative Analysis of Communicable Disease Models,” Mathematical Biosciences, Vol. 28, No. 3-4, 1976, pp. 335-356.

[29] Bony and J. Michel, “Principe du Maximum, Inégalite de Harnack et Unicitédu Problème de Cauchy Pour les Opérateurs Elliptiques Dégénérés,” Annales de l’institut Fourier, Vol. 19, No. 1, 1969, pp. 277-304.

[30] M. Quincampoix, “Differential Inclusions and Target Problems,” SIAM Journal on Control and Optimization, Vol. 30, No. 2, 1992, pp. 324-335.

[31] M. Vidyasagar, “Decomposition Techniques for Large Scale Systems with Nonadditive Interactions: Stability and Stabilizability,” IEEE Transactions on Automatic Control, Vol. 25, No. 4, 1980, pp. 773-779.

[32] P. Van Den Driessche and J. Watmough, “Reproduction Numbers and Subthreshold Endemic Equilibria for Com partmental Models of Disease Transmission,” Mathe matical Biosciences, Vol. 180, No. 1-2, 2002, pp. 29-48.

[33] O. Diekmann, J. A. P. Heesterbeek and J. A. J. Metz, “On the Definition and the Computation of the Basic Repro duction Ratio R0 in Models for Infectious Diseases in Heterogeneous Populations,” Journal of Mathematical Biology, Vol. 28, No. 4, 1990, pp. 365-382.

[34] J. K. Hale, “Ordinary Differential Equations,” Krieger, 1980.

[35] J. P. LaSalle, “The Stability of Dynamical Systems,” Society for Industrial and Applied Mathematics, Phila delphia, 1976.

[36] J. P. LaSalle, “Stability Theory for Ordinary Differential Equations,” Journal of Differential Equations, Vol. 4, No. 1, 1968, pp. 57-65.

[37] J. A. Yorke, H. W. Hethcote and A. Nold, “Dynamics and Control of the Transmission of Gonorrhea,” Sexually Transmitted Diseases, Vol. 5, No. 2, 1978, pp. 51-56.

[38] C. C. McCluskey, “Global Stability for an SIR Epidemic Model with Delay and Nonlinear Incidence,” Nonlinear Analysis: Real World Applications, Vol. 11, No. 4, 2010, pp. 3106-3109.